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I decided to follow a recent trend and ask a question about logarithmic integrals :)

Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$

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  • $\begingroup$ See this sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf and this math.stackexchange.com/questions/155/… $\endgroup$ – math_man Sep 16 '14 at 18:14
  • $\begingroup$ @math_man There are some general results about non-existence of indefinite integrals in certain terms for some classes of functions, but AFAIK there is no such general theory for definite integrals. $\endgroup$ – Vladimir Reshetnikov Sep 16 '14 at 19:02
  • $\begingroup$ By writing $\log(1+x) = \log(1-x^2) - \log(1-x)$, we can express the integral in terms of derivatives of the Beta function. It is tedious and not very interesting to calculate the result, but it certainly exists in terms of polygamma functions. $\endgroup$ – user111187 Sep 16 '14 at 20:03
  • $\begingroup$ @user111187: I don't think so. $\endgroup$ – Lucian Sep 16 '14 at 23:36
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    $\begingroup$ @Lucian Yes, amd also this, this, this, this and this. $\endgroup$ – Nik Z. Sep 19 '14 at 16:51
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This integral is equal to $$ -4\big( \zeta(-3,-1,-1,-1) +\zeta(-3,-1,1,-1) +\zeta(-3,1,-1,1) +\zeta(3,-1,-1,-1) +\zeta(3,-1,1,-1) +\zeta(3,1,-1,1) \big) $$ in terms of the multiple zeta function, which can also be simplified to $$ 2\zeta(-5,-1)-2\zeta(-5,1)+2\zeta(5,-1)+{\textstyle\frac32}\zeta(5,1)+4\zeta(-3,1,1,1), $$ of which only $$ \begin{aligned} \zeta(5,1) &= {\textstyle\frac34}\zeta(6)-{\textstyle\frac12}\zeta(3)^2 \\ \zeta(5,-1) &= {\textstyle\frac{111}{64}} \zeta (6)-{\textstyle\frac{9}{32}} \zeta (3)^2-{\textstyle\frac{31}{16}} \zeta (5) \log (2) \end{aligned} $$ have a known closed form (see also this article about Euler sums, and also Euler Sums and Contour Integral Representations by Philippe Flajolet and Bruno Salvy).

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  • $\begingroup$ Could you express it in terms of non-multiple zeta and polylogarithms? $\endgroup$ – Vladimir Reshetnikov Sep 21 '14 at 1:03
  • $\begingroup$ @VladimirReshetnikov No, I tried weight-6 combinations of zeta functions, and logs and polylogs of arguments $\frac{u}{v}$, $1\leq u\leq 4$. But I'm not confident an expression doesn't exist. $\endgroup$ – Kirill Sep 21 '14 at 1:05
  • $\begingroup$ Perhaps $\zeta(-5,-1)-\zeta(-5,+1)$ could possess a closed form even if neither term separately is known to have one. $\endgroup$ – Lucian Apr 7 '17 at 21:36
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There is no closed form for this integral as the answer involves $\sum_{n=1}^\infty\frac{H_n}{n^52^n}$ and $\sum_{n=1}^\infty\frac{(-1)^nH_n}{n^5}$ which have no known closed form and here is how I found them:

Let $I$ denotes our integral $\displaystyle \int_0^1\frac{\ln x\ln^2(1-x)\ln^2(1+x)}{x}\ dx$

Using the algebraic identity

$$12a^2b^2=(a+b)^4+(a-b)^4-2a^4-2b^4$$ and by letting $a=\ln(1-x)$ and $b=\ln(1+x)$ we can write our integral :

$$\small{12I=\underbrace{\int_0^1\frac{\ln x\ln^4(1-x^2)}{x}}_{1-x^2\mapsto x}+\underbrace{\int_0^1\frac{\ln x\ln^4\left(\frac{1-x}{1+x}\right)}{x}}_{\frac{1-x}{1+x}\mapsto x}-2\underbrace{\int_0^1\frac{\ln x\ln^4(1-x)}{x}}_{1-x\mapsto x}\ dx-2\int_0^1\frac{\ln x\ln^4(1+x)}{x}\ dx}$$

$$12I=-\frac74\underbrace{\int_0^1\frac{\ln(1-x)\ln^4x}{1-x}\ dx}_{K}+2\underbrace{\int_0^1\frac{\ln\left(\frac{1-x}{1+x}\right)\ln^4x}{1-x^2}\ dx}_{J}-2\underbrace{\int_0^1\frac{\ln x\ln^4(1+x)}{x}\ dx}_{M}$$


$$K=\int_0^1\frac{\ln(1-x)\ln^4x}{1-x}\ dx=-\sum_{n=1}^\infty H_n\int_0^1x^n\ln^4x\ dx\\ =-24\sum_{n=1}^\infty\frac{H_n}{(n+1)^5}=-24\sum_{n=1}^\infty\frac{H_n}{n^5}+24\zeta(6)=\boxed{12\zeta^2(3)-18\zeta(6)}$$


To evaluate $J$ we are going to use the identity

$$\frac{1}{1-x^2}\ln\left(\frac{1-x}{1+x}\right)=\sum_{n=1}^{\infty}\left(H_n-2H_{2n}\right)x^{2n-1}$$

$$J=\int_0^1\frac{\ln\left(\frac{1-x}{1+x}\right)\ln^4x}{1-x^2}\ dx=\sum_{n=1}^{\infty}\left(H_n-2H_{2n}\right)\int_0^1x^{2n-1}\ln^4x\ dx\\ \sum_{n=1}^{\infty}\left(H_n-2H_{2n}\right)\left(\frac{3}{4n^5}\right)=-\frac{93}{4}\sum_{n=1}^\infty\frac{H_n}{n^5}-24\sum_{n=1}^\infty\frac{(-1)^nH_n}{n^5}\\ =\boxed{\frac{93}{8}\zeta^2(3)-\frac{651}{16}\zeta(6)-24\sum_{n=1}^\infty\frac{(-1)^nH_n}{n^5}}$$


I managed to simplify $M$ here

$$M=-120\operatorname{Li}_6\left(\frac12\right)-72\ln2\operatorname{Li}_5\left(\frac12\right)-24\ln^22\operatorname{Li}_4\left(\frac12\right)+78\zeta(6)+\frac34\ln2\zeta(5)-\frac32\ln^22\zeta(4)-3\ln^32\zeta(3)+2\ln^42\zeta(2)+12\zeta^2(3)-12\ln2\zeta(2)\zeta(3)-\frac{17}{30}\ln^62+24\sum_{n=1}^\infty\frac{H_n}{n^52^n}$$


Combining the results of $K$, $J$ and $M$ we get

$$I=20\operatorname{Li}_6\left(\frac12\right)+12\ln2\operatorname{Li}_5\left(\frac12\right)+4\ln^22\operatorname{Li}_4\left(\frac12\right)-\frac{549}{32}\zeta(6) -\frac18\ln2\zeta(5)+\frac14\ln^22\zeta(4)\\ +\frac12\ln^32\zeta(3)-\frac13\ln^42\zeta(2)-\frac{29}{16}\zeta^2(3)+2\ln2\zeta(2)\zeta(3)\\ +\frac{17}{180}\ln^62-4\sum_{n=1}^\infty\frac{H_n}{n^52^n}-4\sum_{n=1}^\infty\frac{(-1)^nH_n}{n^5}$$

and here we see the two sums appeared and because their numerical values (given by wolfram) are different so unfortunately they don't cancel each other out. So the integral $I$ has no closed form.

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